60.1.41 problem Problem 55

Internal problem ID [15169]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 1, First-Order Differential Equations. Problems page 88
Problem number : Problem 55
Date solved : Thursday, October 02, 2025 at 10:06:29 AM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} 3 y^{2}-x +2 y \left (y^{2}-3 x \right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.099 (sec). Leaf size: 101

ode:=3*y(x)^2-x+2*y(x)*(y(x)^2-3*x)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\sqrt {-2 \sqrt {c_1 \left (c_1 -8 x \right )}+2 c_1 -4 x}}{2} \\ y &= \frac {\sqrt {-2 \sqrt {c_1 \left (c_1 -8 x \right )}+2 c_1 -4 x}}{2} \\ y &= -\frac {\sqrt {2 \sqrt {c_1 \left (c_1 -8 x \right )}+2 c_1 -4 x}}{2} \\ y &= \frac {\sqrt {2 \sqrt {c_1 \left (c_1 -8 x \right )}+2 c_1 -4 x}}{2} \\ \end{align*}

✓ Mathematica. Time used: 9.001 (sec). Leaf size: 185

ode=(3*y[x]^2-x)+(2*y[x])*(y[x]^2-3*x)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\sqrt {-2 x-e^{\frac {c_1}{2}} \sqrt {8 x+e^{c_1}}-e^{c_1}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-2 x-e^{\frac {c_1}{2}} \sqrt {8 x+e^{c_1}}-e^{c_1}}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {-2 x+e^{\frac {c_1}{2}} \sqrt {8 x+e^{c_1}}-e^{c_1}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-2 x+e^{\frac {c_1}{2}} \sqrt {8 x+e^{c_1}}-e^{c_1}}}{\sqrt {2}} \end{align*}

✓ Sympy. Time used: 8.921 (sec). Leaf size: 160

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x + (-6*x + 2*y(x)**2)*y(x)*Derivative(y(x), x) + 3*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 x e^{2 C_{1}} - \sqrt {- 8 x e^{2 C_{1}} + 1} + 1} e^{- C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 x e^{2 C_{1}} + \sqrt {- 8 x e^{2 C_{1}} + 1} + 1} e^{- C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 x e^{2 C_{1}} + \sqrt {- 8 x e^{2 C_{1}} + 1} + 1} e^{- C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 x e^{2 C_{1}} - \sqrt {- 8 x e^{2 C_{1}} + 1} + 1} e^{- C_{1}}}{2}\right ] \]